Heegaard 分解的双曲性及距离不下降的把柄添加的一些问题

By Thurston's Geometrization conjecture and Hempel's theorem, if a 3-manifold admits a distance at least 3 Heegaard splitting, then it is hyperbolic. But for a 3-manifold admitting only distance at most 2 Heegaard splittings, it could be hyperbolic, a Seifert fiber space, or admits none of Thurston's eight geometries. Thus it is interesting to determine a hyperbolic 3-manifold admitting a distance 2 Heegaard splitting. It is known that a handle addition or Dehn filling doesn't increase the Heegaard distance. Thus to give a description of all distance non degenerate handle additions or Dehn fillings is important in studying a Heegaard splitting..This project will concern on geometry of the curve complex , topological properties of a compression body. By using them, it is expected to give a condition to determine the geometry of a distance 2 Heegaard splitting and a description of all distance non degenerate handle additions or Dehn fillings.

由Thurston 几何化猜想以及Hempel的结果可知，给定一个三维流形，如果它的某一个Heegaard分解的距离至少为3，那么它是一个双曲三维流形。而如果该三维流形仅仅只有距离不超过2的Heegaard 分解，那么它有可能是双曲的，可能是一个Seifert纤维空间，也可能是非八种几何之一。因此给出该流形双曲性的一个判定条件为大家关心。一个Heegaard分解的把柄添加或者Dehn 填补所产生的新Heegaard分解的距离不会变大。因此给出保持Heegaard距离不下降的把柄添加或者Dehn填补的一个描述也是Heegaard 分解研究的重点问题。.本项目将结合曲线复形的几何性质，压缩体的拓扑性质及三维流形的几何性质，研究上述问题。本项目希望得到距离为2的Heegaard分解的双曲性的一个判定条件；给出保持Heegaard距离不下降的把柄添加或者Dehn填补的一个描述。

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